Gauge Invariant Observable

For the tachyon vacuum solution g₀, the GIO becomes Wpg0,Vq “Tr_VrΨ0s

“Tr_V

„

pQpBcq ´cq 1 1´K

ȷ

“ ´Tr_V

„ c 1

1´K ȷ

“0´A0pVq. (7.2.38)

For the trivial solution g₁, it becomes

Wpg1,Vq “ 0“A0pVq ´A0pVq. (7.2.39) For our solution g₂, it is calculated as

limϵÑ0JWpg₂,VqKϵ “lim

ϵÑ0Tr_VrJΨ₂Kϵs “lim

ϵÑ0Tr_V

„ˆ

Bγ²`cB K_ϵ² 1´K_ϵc

˙ 1

´K_ϵ ȷ

“lim

ϵÑ0Tr_V

„

cB K_ϵ 1´Kϵ

Bc 1

´Kϵ

ȷ

“ ´Tr_V

„

cB 1 1´KBc

ȷ

“Tr_V

„

BQc 1 1´K

ȷ

“Tr_V

„ c K

1´K ȷ

“Tr_V

„ c 1

1´K ȷ

“2A0pVq ´A0pVq. (7.2.40) Here, we used

Tr_VrBγ²fpKqs “0, (7.2.41) which follows from the ϕ-momentum conservation, and also

Tr_Vrφ₁Qφ₂s “ ´p´q^ϵpφ1qTr_VrQφ₁¨φ₂s, (7.2.42) since V is on-shell. Our solution satisfies a needed property of the double brane solution.

Namely, the value of the GIO of our solution (7.2.40) is larger than that of the tachyon vacuum solution by the value which seems to be consistent with the existence of two D9-branes. When we choose the vertex operator V to be the time-like component of the graviton

VG :“ i

2πpξ`ξqc˜˜ ce^´ϕe^´ϕ˜ψ⁰ψ˜⁰, (7.2.43)

then the disk amplitude corresponds to the energy of the solution:

A0pVGq “ 1 2πi

i

2πxξ`ξ˜y^ξ_UHPxe^´ϕpiqe^´ϕp´iq y^ϕ_S2

ˆ xcpiqcp´iqp´1

2iq^´1cp0q y^bc_S2xψ⁰piqψ⁰p´iq y^ma_S2

“ 1 2πi

i

2πp´2q 1

i`iˆ p2iqpi`iqpi´0qp´i´0q ˆ η⁰⁰ i`i

“ 1 2π²

“T₉. (7.2.44)

Then, we obtain

Wpg0,Vgq “ Epg0q, Wpg₁,Vgq “ Epg₀q `T₉,

limϵÑ0JWpg₂,VgqKϵ “Epg₀q `2T₉. (7.2.45)

Chapter 8 Conclusion

We constructed three types of the new multiple-brane solutions by using singular gauge transformations in different theories.

First, we constructed candidates for the solution of the EOM in the bosonic cubic SFT. They were obtained by performing the singular gauge transformation whose gauge parameter is U₁^´1 for the EM soltuion; the number of times of the singular gauge trans-formation is equal to n. Since, in general, these candidates include the singular string field 1{K, we adopted the K_ϵ-regularization and checked the EOM in the strong sense (EOMS). After this checking, we realize that the only candidate which satisfies the EOMS is the one for n “ 1. We evaluated the energy of our solution, and then we found that the singular gauge transformation increases the energy by the value of the tension of the D25-brane. We also calculated the tachyon profile, by using the Neumann–Dirichlet twist operators as the boundary condition changing operators. The plotted figure shows that our solution describes the D24-brane on the D25-brane; these D-branes are originated from the EM solution and the gauge transformation, respectively. This result gives a support for that the singular gauge transformation U1´1 creates the D25-brane in this case.

Second, in the modified cubic superstring field theory, we constructed a candidate for the solution of the EOM by performing the singular gauge transformation for the tachyon vacuum solution three times. Here we took the singular gauge parameter U_1{2^´1. As our first solution, this solution includes the singular string field 1{G, then we introduced the Gϵ-regularization, and we checked that the solution satisfies the EOMS. We also evaluated the energy, and the result is expected one, i.e., the energy of our solution is increased from the energy of the tachyon vacuum solution by 3/2 times the tension of the D9-brane. Since Ψ₂does not satisfy the EOMS, a pure-gauge-form string fieldU_1{2²QU_1{2^´2, which is gauge equivalent toΨ₂, does not satisfy the EOMS. Therefore, we did not consider further gauge transformations with U_1{2^´1.

Third, we constructed a candidate for the double-brane solution by performing the singular gauge transformation from the tachyon vacuum in the Berkovits’ superstring field theory. We gave the integral form of the energy of the candidate but did not reach the final result because the integral is complicated and lengthy. We also discussed the regularization in this theory. We gave another support that the candidate is the

double-brane solution. This was given by evaluating the GIO, and we found the value which is consistent with the double-brane solution.

Let us give some comments regarding future directions. Since we have not yet com-pleted the computation of the energy of the double-brane solution in Berkovits’ SFT, to accomplish this task should be the important future work. Since in this thesis, we mainly evaluated the solutions by using their energies and EOMSs, it would be interesting to investigate other properties of the solutions. Regarding such a direction, we would say that the D-brane charge should be an interesting quantity to be studied. However, so-lutions studied in this thesis does not include any stable BPS D-branes. What is more, since the D-brane construction studied in this thesis is based on the singular gauge trans-formations connecting the unstable perturbative vacuum and the tachyon vacuum, It is not clear whether this method is, in any sense, useful to construct stable BPS D-branes with charges. These issues including the investigations of further method of constructing D-branes are important and interesting future directions.

Acknowledgements

First of all, I would like to thank my supervisor, Akitsugu Miwa, for his guidance and all his help. Without his help, this thesis must be unfinished for ever more. I am very grateful to Shigefumi Naka, Shinichi Deguchi and Takeshi Nihei for a careful reading of the manuscript. I am also grateful Satoshi Ohya, Takashi Mishima, Kouichirou Umetsu, Takayuki Enari, Naohiro Kanda, Satoshi Okano and the other members of Nihon Univer-sity for useful discussions. Finally, I want to thank my family for their support through my years of study.

Appendix A

Correlators and Formulae in the Bosonic Cubic String Field Theory

We give formulae of the correlators [15] in the sliver frame. The basic formula is TrrcΩ^t1cΩ^t2cΩ^t3s “ ´

ˆL π

˙3

sinθ_t1sinθ_t2sinθ_t3, (A.1) where Lis the circumference of the sliver, now L“t1`t2`t3, andθt:“ ^πt_L. This can be derived from the three point function of cpzq

xcpz₁qcpz₂qcpz₃q y^bc_UHP“z₁₂z₁₃z₂₃, pz_ij :“z_i´z_jq (A.2) as follows:

TrrcΩ^t1cΩ^t2cΩ^t3s “ xcp0qcpt₁qcpt₁`t₂q y^bc_CL

“ xf_LÑ2˝cp0qf_LÑ2˝cpt₁qf_LÑ2˝cpt₁`t₂q y^bc_C2

“ ˆ2

L

˙´3

xcp0qc ˆ2t₁

L

˙ c

ˆ2pt₁`t₂q L

˙ y^bc_C2

“ ˆ2

L

˙´3

xf_s^´1 ˝cp0qf_s^´1˝c ˆ2t₁

L

˙

f_s^´1 ˝c

ˆ2pt₁`t₂q L

˙ y^bc_UHP

“ ˆ2

L

˙´3ˆ π 2

˙´3

cos²θ_t1cos²θ_{t1`t2</sub>xcp0qcptanθ<sub>t1</sub>qcptanθ<sub>t1`t2}q y^bc_UHP

“ ˆL

π

˙3

cos²θ_t1cos²θ_t1`t2

ˆ p0´tanθ_t1qp0´tanθ_{t1`t2</sub>qptanθ<sub>t1</sub> ´tanθ<sub>t1`t2}q

“ ˆL

π

˙3

sinθ_t1sinθ_{t1`t2</sub>psinθ<sub>t1</sub>cosθ<sub>t1`t2} ´sinθ_t1`t2cosθ_t1q

“ ´ ˆL

π

˙3

sinθ_t1sinθ_t1`t2sinθ_t2

“ ´

ˆL˙3

sinθ_t sinθ_t sinθ_t , (A.3)

where f_s^´1pξq “ tan^πξ₂, Bξf_s^´1pξq “ ^π₂ ¹

cos^{2 πξ}₂ , sinθ_L´t“sinθ_t. Next we give the correlator with a string field B and four c’s:

TrrBcΩ^t1cΩ^t2cΩ^t3cΩ^t4s “ ´t₁ L

ˆL π

˙3

sinθt1`t2sinθt3sinθt4

` t<sub>1</sub>`t₂ L

ˆL π

˙3

sinθt1sinθt2`t3sinθt4

´ t₁`t<sub>2</sub>`t₃ L

ˆL π

˙3

sinθt1sinθt2sinθt3`t4 (A.4)

“ ´L²

4π³pt₃sin 2θ_t1 ´ pt₂`t<sub>3</sub>qsin 2θ<sub>t1`t2</sub> `t<sub>2</sub>sin 2θ<sub>t1`t2`t3</sub>

`t<sub>1</sub>sin 2θ<sub>t3</sub> ´ pt<sub>1</sub>`t₂qsin 2θ<sub>t2`t3</sub> ` pt₁`t<sub>2</sub>`t₃qsin 2θ_t2q.

(A.5) To derive this, we consider the following correlator:

TrrBcΩ^t1cΩ^t2cΩ^t3cΩ^t4s “ x ż

Ó0

dz

2πibpzqcp0qcpt₁qcpt₁`t<sub>2</sub>qcpt<sub>1</sub>`t₂`t₃q y^bc_CL. (A.6) We start with the following relation:

x ż

Ó´0

dz

2πizbpzqcp0qcpt₁qcpt₁`t<sub>2</sub>qcpt<sub>1</sub>`t₂`t₃q y^bc_CL

“ x ż

Ó´0

dz

2πizbpzqcp0qcpt₁qcpt₁`t<sub>2</sub>qcpt<sub>1</sub>`t₂`t₃q y^bc_CL

` x ˆż

Ó`0

` ż

Ò`0

˙ dz

2πizbpzqcp0qcpt₁qcpt₁`t<sub>2</sub>qcpt<sub>1</sub>`t₂`t₃q y^bc_CL

“ x ˆż

Ó´0

` ż

Ò`0

˙ dz

2πizbpzqcp0qcpt₁qcpt₁`t<sub>2</sub>qcpt<sub>1</sub>`t₂`t₃q y^bc_CL

` p´q^ϵpbqϵpcqxcp0q ż

Ó`0

dz

2πizbpzqcpt1qcpt1`t2qcpt1`t2`t3q y^bc_CL

“ x ˆ¿

0

dz

2πizbpzqcp0q

˙

cpt1qcpt1`t2qcpt1`t2`t3q y^bc_CL

´ xcp0q ż

Ó`0

dz

2πizbpzqcpt₁qcpt₁`t<sub>2</sub>qcpt<sub>1</sub>`t₂`t<sub>3</sub>q y<sup>bc</sup><sub>CL</sub>. (A.7) After repeated uses of the similar relations around z “t<sub>1</sub>,z “t<sub>1</sub>`t₂ and z “t₁`t<sub>2</sub>`t₃, we obtain

(A.7)“ x ˆ¿

0

dz

2πizbpzqcp0q

˙

cpt₁qcpt₁`t<sub>2</sub>qcpt<sub>1</sub>`t₂`t₃q y^bc_CL

´ xcp0q ˆ¿

t1

dz

2πizbpzqcpt₁q

˙

cpt₁`t<sub>2</sub>qcpt<sub>1</sub>`t₂`t₃q y^bc_CL

` xcp0qcpt₁q ˆ¿

t1`t2

dz

2πizbpzqcpt₁ `t₂q

˙

cpt₁`t<sub>2</sub>`t₃q y^bc_CL

´ xcp0qcpt₁qcpt₁`t₂q ˆ¿

t1`t2`t3

dz

2πizbpzqcpt₁`t<sub>2</sub>`t₃q

˙ y^bc_CL

` xcp0qcpt1qcpt1`t2qcpt1`t2`t3q ż

ÓL´0

dz

2πizbpzq y^bc_CL

“ x p0q ¨cpt1qcpt1`t2qcpt1`t2`t3q y^bc_CL

´ xcp0q ¨ pt₁q ¨cpt₁`t<sub>2</sub>qcpt<sub>1</sub> `t₂ `t₃q y^bc_CL

` xcp0qcpt<sub>1</sub>q ¨ pt<sub>1</sub>`t₂q ¨cpt₁ `t<sub>2</sub> `t₃q y^bc_CL

´ xcp0qcpt1qcpt1`t2q ¨ pt1`t2`t3q y^bc_CL

` x ż

Ó´0

dz

2πipz`Lqbpzqcp0qcpt<sub>1</sub>qcpt<sub>1</sub> `t₂qcpt₁`t<sub>2</sub>`t₃q y^bc_CL, (A.8) where we used the periodicity z »z `L of the cylinder. Then, we obtain the following correlator:

x ż

Ó´0

dz

2πizbpzqcp0qcpt1qcpt1`t2qcpt1`t2`t3q y^bc_CL

“ ´t₁xcp0qcpt₁`t<sub>2</sub>qcpt<sub>1</sub> `t₂ `t₃qy^bc_CL

` pt<sub>1</sub>`t₂qxcp0qcpt₁qcpt₁`t<sub>2</sub>`t₃q y^bc_CL

´ pt₁`t<sub>2</sub>`t₃qxcp0qcpt₁qcpt₁`t₂q y^bc_CL

` x ż

Ó´0

dz

2πipz`Lqbpzqcp0qcpt<sub>1</sub>qcpt<sub>1</sub>`t₂qcpt₁ `t<sub>2</sub> `t₃q y^bc_CL. (A.9) Therefore, (A.4) is obtained:

TrrBcΩ^t1cΩ^t2cΩ^t3cΩ^t4s “ t₁

LTrrcΩ^t1`t2cΩ^t3cΩ^t4s

´ t1`t2

L TrrcΩ^t1cΩ^t2`t3cΩ^t4s

` t<sub>1</sub>`t₂`t₃

L TrrcΩ^t1cΩ^t2cΩ^t3`t4s

“ ´t₁ L

ˆL π

˙3

sinθ_t1`t2sinθ_t3sinθ_t4

` t<sub>1</sub>`t₂ L

ˆL π

˙3

sinθ_t1sinθ_t2`t3sinθ_t4

´ t₁`t<sub>2</sub>`t₃ L

ˆL π

˙3

sinθ_t1sinθ_t2sinθ_t3`t4. (A.10) Furthermore, by using

sinx₁sinx₂sinx₃ “ 1 4

`´sinpx<sub>1</sub>`x₂`x<sub>3</sub>q `sinpx₁`x₂´x₃q

`sinpx<sub>1</sub>´x<sub>2</sub>`x₃q ´sinpx₁´x₂´x₃q˘

, (A.11)

which can be derived from

sinpx1`x2`x3q “sinx1cosx2cosx3´sinx1sinx2sinx3

`cosx<sub>1</sub>sinx<sub>2</sub>cosx<sub>3</sub>`cosx₁cosx₂sinx₃, (A.12) we can rewrite it:

(A.10)“ ´t1

L ˆL

π

˙3

1 4

`´sinθ<sub>pt1`t2q`t3`t4</sub> `sinθ<sub>pt1`t2q`t3´t4</sub>

`sinθ<sub>pt1`t2q´t3`t4</sub> ´sinθ<sub>pt1`t2q´t3´t4</sub>q

`t<sub>1</sub>`t₂ L

ˆL π

˙3

1 4

`´sinθ<sub>t1`pt2`t3q`t4</sub> `sinθ<sub>t1`pt2`t3q´t4</sub>

`sinθ<sub>t1´pt2`t3q`t4</sub> ´sinθ<sub>t1´pt2`t3q´t4</sub>˘

´t₁`t<sub>2</sub>`t₃ L

ˆL π

˙3

1 4

`´sinθ<sub>t1`t2`pt3`t4q</sub>`sinθ<sub>t1`t2´pt3`t4q</sub>

`sinθ<sub>t1´t2`pt3`t4q</sub>´sinθ<sub>t1´t2´pt3`t4q</sub>˘

“ ´ L²

4π³pt₃sin 2θ_t1 ´ pt₂`t<sub>3</sub>qsin 2θ<sub>t1`t2</sub> `t<sub>2</sub>sin 2θ<sub>t1`t2`t3</sub>

`t<sub>1</sub>sin 2θ<sub>t3</sub> ´ pt<sub>1</sub>`t₂qsin 2θ<sub>t2`t3</sub> ` pt₁`t<sub>2</sub>`t₃qsin 2θ_t2q, (A.13) where t₄ “L´ pt₁`t<sub>2</sub>`t₃q and sinpπ´θq “ sinθ. For simplicity, we define

Bccccrt1, t2, t3, t4s:“TrrBcΩ^t1cΩ^t2cΩ^t3cΩ^t4s. (A.14) By using this notation,

Bccccrt₁, t₂, t₃, t₄s “ ´ L²

4π³pt₃sin 2θ_t1 ´ pt₂`t<sub>3</sub>qsin 2θ<sub>t1`t2</sub> `t<sub>2</sub>sin 2θ<sub>t1`t2`t3</sub>

`t<sub>1</sub>sin 2θ<sub>t3</sub> ´ pt<sub>1</sub>`t₂qsin 2θ<sub>t2`t3</sub> ` pt₁ `t<sub>2</sub> `t₃qsin 2θ_t2q.

(A.15) We further define

Bcdddrt₁, t₂, t₃s:“TrrBcBcΩ^t1BcΩ^t2BcΩ^t3s, Bccddrt1, t2, t3, t4s:“TrrBcΩ^t1cΩ^t2BcΩ^t3BcΩ^t4s,

Bcccdrt₁, t₂, t₃, t₄s:“TrrBcΩ^t1cΩ^t2cΩ^t3BcΩ^t4s,

Bccdcrt₁, t₂, t₃, t₄s:“TrrBcΩ^t1cΩ^t2BcΩ^t3cΩ^t4s, (A.16) and by using

K “ BypΩ^yq|yÑ0, (A.17)

we obtain

Bcdddrt₁, t₂, t₃s “lim

yÑ0By¹Bccddry, t₁, t₂, t₃s

“ lim

y,y¹Ñ0ByBy¹

!

Bcccdry, t₁`y<sup>1</sup>, t<sub>2</sub>, t<sub>3</sub>s ´Bcccdry, t<sub>1</sub>, t<sub>2</sub>`y¹, t₃s )

“ lim

y,y¹,y²Ñ0ByBy¹By²

!

Bccccry, t₁`y<sup>1</sup>, t<sub>2</sub>`y², t₃s ´Bccccry, t₁`y<sup>1</sup>, t<sub>2</sub>, t<sub>3</sub>`y²s

´Bccccry, t₁, t₂`y<sup>1</sup>`y², t₃s `Bccccry, t<sub>1</sub>, t<sub>2</sub>`y¹, t₃`y²s )

“ ´1

πpsin 2θ_t2 `sin 2θ<sub>t3</sub> ´sin 2θ<sub>t2`t3</sub>q (A.18) Similary,

Bccdcrt₁, t₂, t₃, t₄s

“ L² 4π³

ˆ ˆ2π L

˙

`´ pt1`t2`t3qcos 2θt2 ` pt2`t3qcos 2θt1`t2 `t1cos 2θt3

˘

`sin 2θt1 `sin 2θt2`t3 ´sin 2θt1`t2`t3

˙

. (A.19)

Appendix B

KBcGγ Algebra

We summarize the derivations of the KBcGγ algebras.

• Bˆ² “cˆ² “0

B²bI “c²bI “0 (B.1)

• tB,ˆ ˆcu “1

tB,ˆ cu “ pBcˆ `cBq bI₂ ÑCFT

ż

Ó´0

dz

2πibpzqcp0q ` ż

Ó`0

dz

2πicp0qbpzq

“ ˆż

Ó´0

` ż

Ò`0

˙ dz

2πibpzqcp0q

“

¿

0

dz

2πibpzqcp0q

“1

ÑSFT 1bI₂ “ˆ1 :“1 (B.2)

• tˆγ,Bu “ tˆˆ γ,cu “ˆ 0

tˆγ,ˆcu “ pγc´cγq biσ₂σ₃ “0, tˆγ,Bu “ pγBˆ ´Bγq biσ₂σ₃ “0. (B.3) 7 rγ, Bs “ rγ, cs “0.

• δˆGˆ“2 ˆK

δˆGˆ “ tG,ˆ Gu “ pGGˆ `GGq bI₂ ÑCFT

ż

Ó0

dz 2πi

ż

Ó`0

dw

2πiGpzqGpwq ` ż

Ó´0

dw 2πi

ż

Ó0

dz

2πiGpwqGpzq

“ ż

Ó0

dz 2πi

ˆż

Ó´0

` ż

Ò`0

˙ dw

2πiGpwqGpzq

“ ż

Ó0

dz 2πi

¿

z

dw 2πi

2Tpwq w´z

“2 ż

Ó0

dz 2πiTpzq

ÑSFT 2KbI₂ “2 ˆK (B.4)

• δˆˆc“2ˆγ

rG,ˆ ˆcs “ pGc`cGq bσ₁σ₃ ÑCFT

ż

Ó´0

dz

2πiGpzqcp0q ` ż

Ó`0

dz

2πicp0qGpzq

“ ˆż

Ó´0

` ż

Ò`0

˙ dz

2πiGpzqcp0q

“

¿

0

dz 2πi

´2γp0q z

“ ´2γp0q

ÑSFT ´2γb ´iσ₂ “2ˆγ (B.5)

• δˆˆγ “Bˆˆc{2

tG,ˆ γu “ pGγˆ ´γGq bσ1iσ2 ÑCFT

ż

Ó´0

dz

2πiGpzqγp0q ´ ż

Ó`0

dz

2πiγp0qGpzq

“ ˆż

Ó´0

` ż

Ò`0

˙ dz

2πiGpzqγp0q

“

¿

0

dz 2πi

´Bcp0q 2z

“ ´ 1 2

ˆż

Ó0´

dz

2πiTpzqcp0q ´ ż

Ó0`

cp0qTpzq

˙

ÑSFT ´ 1

2Bcb p´σ₃q “ 1 2

ˆBcˆ (B.6)

• δˆˆγ² “2ˆδˆγ¨ˆγ

rG,ˆ γˆ²s “ tG,ˆ ˆγuˆγ´γtˆ G,ˆ γu “ˆ Bˆˆcˆγ (B.7)

• QˆBˆ “Kˆ

QˆBˆ “QBbI₂ ÑCFT

¿

w

dz 2πi

ż

Ó0

dw

2πij_Bpzqbpwq

“ ż

Ó0

dw 2πiTpwq

ÑSFT K bI2 “Kˆ (B.8)

• rK,ˆ Bˆs “0

0“Qpˆ Bˆ¨Bqˆ

“QˆBˆ¨Bˆ´BˆQˆBˆ

“KˆBˆ´BˆKˆ (B.9)

• QˆKˆ “0

QˆKˆ “Qˆ²Bˆ “0 (B.10)

• Qˆˆc“cˆˆBˆc`γˆ²

Qˆˆc“QcbI₂ ÑCFT

¿

0

dz

2πij_Bpzqcp0q

“cBc´γ²p0q

ÑSFT cBcbI₂`γ²b piσ₂q²

“ˆcˆBcˆ`ˆγ² (B.11)

• QˆGˆ “0

QˆGˆ “QGbiσ₂ “0 (B.12)

• Qˆˆγ “ˆcBˆˆγ´Bˆˆcˆγ{2

Qˆˆγ “Qγbσ₃iσ₂ ÑCFT

¿

0

dz

2πij_Bpzqγp0q

“pcBγ´ 1

2Bcγqp0q ÑSFT pcBγ´ 1

2Bcγq bσ₃iσ₂

“ˆcˆBˆγ´ 1 2

Bˆˆcˆγ (B.13)

Appendix C

Correlators in Modified Cubic String Field Theory

To calculate energy of solutions, we introduce the formulae of the correlators. The non-vanishing correlator in this theory is now normalized as

xcBcB²ce^´2ϕpzqy^gh_UHP“ ´2. (C.1) In the correlator, the ghost number is 3, the bc-ghost number is 3, the ϕ momentum is

´2, and the picture is ´2. We derive the basic correlator:

TrY´2rcγΩ^t1γΩ^t2s

“ xY_´2pi8qcp0qγp0qγpt₁q y^gh_CLˆ 1

2Trrσ₃σ₃iσ₂iσ₂s

“ ´xY_´2piqf_s^´1˝f_LÑ2˝cp0qf_s^´1˝f_LÑ2˝γp0qf_s^´1˝f_LÑ2˝γpt₁q y^gh_UHP

“ ´

´2 L

¯´1´¹₂ˆ2´π 2

¯´1´¹₂ˆ2´ 1 cos²θ_t1

¯´¹₂

ˆ xcBξe^´2ϕpiqcBξe^´2ϕp´iqcp0qηe^ϕp0qηe^ϕptanθt1q y^gh_S2

“ ´

´π L

¯´2

cosθ_t1xcpiqcp´iqcp0qy^bc_S² ˆ xe^´2ϕpiqe^´2ϕp´iqe^ϕp0qe^ϕptanθ_t1q y^ϕ_S2

ˆ Bs1Bs2xξps₁qξps₂qηp0qηptanθ_t1q y^ξη_S2|s1“i, s2“´i

“ ´pπ

Lq^´2cosθ_t1pi`iqpi´0qp´i´0q

ˆ pi`iq^´4pi´0q²pi´tanθ_t1q²p´i´0q²p´i´tanθ_t1q²p0´tanθ_t1q^´1 ˆ Bs1Bs2tps1´s2qps1´0q^´1ps1´tanθt1q^´1

¨ ps₂´0q^´1ps₂´tanθ_t1q^´1p0´tanθ_t1qu|s1“i, s2“´i

“ L²cosθ_t1

2π² . (C.2)

Here, the width of the sliver is L, and we used the doubling trick, and OPE:

η »e^´χ, ξ »e^χ, χpzqχp0q „lnz,

ϕpzqϕp0q „ ´lnz. (C.3) Next, we derive the correlator Tr_Y´2rBcΩ^t1cγΩ^t2γΩ^t3s by using the technique used in the derivation of TrrBcΩ^t1cΩ^t2cΩ^t3cΩ^t4s. To derive this, let us consider the following equations

xY_´2pi8q ż

Ó´0

dz

2πizbpzqcp0qcγpt₁qγpt₁`t₂q y^gh_CL

“ xY´2pi8q ˆż

Ó´0

` ż

Ò`0

˙ dz

2πizbpzqcp0qcγpt1qγpt1`t2q y^gh_CL

` xY_´2pi8q ż

Ó`0

dz

2πizbpzqcp0qcγpt₁qγpt₁`t₂q y^gh_CL

“ xY_´2pi8qt

¿

0

dz

2πizbpzqcp0qucγpt₁qγpt₁`t₂q y^gh_CL

´ xY´2pi8q ż

Ó`0

dz

2πicp0qzbpzqcpt1qγpt1qγpt1 `t2q y^gh_CL

“ ´xY_´2pi8q

˜ż

Ót1´0

` ż

Òt1`0

¸ dz

2πicp0qzbpzqcpt₁qγpt₁qγpt₁`t₂q y^gh_CL

´ xY_´2pi8q ż

Ót1`0

dz

2πicp0qzbpzqcpt₁qγpt₁qγpt₁`t₂q y^gh_CL

“ ´xY_´2pi8qcp0qt

¿

t1

dz

2πizbpzqcpt₁quγpt₁qγpt₁`t₂q y^gh_CL

` xY´2pi8q ż

ÓL´0

dz

2πicp0qcpt1qγpt1qγpt1`t2qzbpzq y^gh_C

L

“ ´t₁xY_´2pi8qcp0qγpt₁qγpt₁`t₂q y^gh_CL

` xY_´2pi8q ż

Ó´0

dz

2πipz`Lqbpzqcp0qcpt<sub>1</sub>qγpt<sub>1</sub>qγpt<sub>1</sub>`t₂q y^gh_CL. (C.4) Therefore,

Tr_Y´2rBcΩ^t1cγΩ^t2γΩ^t3s “ xY_´2pi8q ż

Ó´0

dz

2πibpzqcp0qcγpt₁qγpt₁`t₂q y^gh_CL ˆ1

2Trrσ₃σ₃σ₃σ₃iσ₂iσ₂s

“ ´t1

LxY´2pi8qcp0qγpt1qγpt1`t2q y^gh_CL

“ ´t₁Lcosθ_t2

2π² . (C.5)

We define

Bccggrt₁, t₂;Ls:“Tr_Y´2rBcΩ^t1cγΩ^t2γΩ^t3s,

Bcdggrt₁;Ls:“Tr_Y´2rBcBcγΩ^t1γΩ^t2s. (C.6) Formulae for the inner product including Bc,Bc, γ, γ is

Tr_Y´2rBcΩ^t1BcΩ^t2γΩ^t3γΩ^t4s “Tr_Y´2rBprc,Ω^t1s `Ω^t1cqBcΩ^t2γΩ^t3γΩ^t4s

“Tr_Y´2rBcBcΩ^t2γΩ^t3γΩ^t4Ω^t1s

“Bcdggrt₃;Ls. (C.7)

Here, the first term in the first line vanishes because TrrBφs “ TrrB²cφs “ 0, for φ s.t.

rB, φs “0. Similarly,

Tr_Y´2rBcΩ^t1γΩ^t2BcΩ^t3γΩ^t4s “Tr_Y´2rBcγΩ^t2BcΩ^t3γΩ^t4Ω^t1s

“Tr_Y´2rBcΩ^t2BcΩ^t3γΩ^t4Ω^t1γs

“Tr_Y´2rBcBcΩ^t3γΩ^t4Ω^t1γΩ^t2s

“Bcdggrt₄ `t₁;Ls, (C.8)

Tr_Y´2rBcΩ^t1γΩ^t2γΩ^t3BcΩ^t4s “Tr_Y´2rBcΩ^t4Ω^t1γΩ^t2γΩ^t3Bcs

“Tr_Y´2rBcBcΩ^t4Ω^t1γΩ^t2γΩ^t3s

“Bcdggrt2;Ls. (C.9)

These only depend on the width between γ’s and total width L.

Appendix D

Detailed Calculation of the Energy of the Half-brane Solution

We give detailed calculation of the energy of the half-brane solution. We can compute it from the cubic term in the action:

Tr_Y´2rΨ_1{2³s “Tr_Y´2

„

Bγ² ´1

1´GBγ² ´1

1´GBγ² ´1 1´G

ȷ

`3Tr_Y´2

„

Bγ² ´1

1´GBγ² ´1

1´GcBp1´GqGc ´1 1´G

ȷ

`3Tr_Y´2

„

Bγ² ´1

1´GcBp1´GqGc ´1

1´GcBp1´GqGc ´1 1´G

ȷ

`Tr_Y´2

«ˆ

cBp1´GqGc ´1 1´G

˙3ff

“ ´3Tr_Y´2

„

Bγ²Gc 1 1´GcG

ȷ

(D.1)

`Tr_Y´2

«ˆ

cBp1´GqGc ´1 1´G

˙3ff

. (D.2)

The first term (D.1) becomes

(D.1)“ ´3Tr_Y´2rBpBcγ`γ<sup>2</sup>GqGcp1`GqΩ_xcs

“ ´3

2Tr_Y´2rδpcΩ_xcBBcγqs ´3Tr_Y´2rBγ²KcΩ_xcs

“6Tr_Y´2rBcBcγ²Ω_xs ´3Tr_Y´2rBcΩ_xcγ²Ks

“ ż8

0

dx e^´x1 ˆ

6¨Bcdggr0;x₁s ´3 lim

yÑ0By¨Bccggrx₁,0;x₁`ys

˙

“ 3

2π², (D.3)

where we used the following equation:

1

1´G “ 1`G

p1´Gqp1`Gq “ p1`GqΩ_x, (D.4) and we define Ω_x :“ _1´K¹ . The second term (D.2) becomes

(D.2)“Tr_Y´2

„

cBp1´GqGc ´1

1´GcBp1´GqGc ´1

1´GcBp1´GqGc ´1 1´G

ȷ

“Tr_Y´2” cBG`

r1´G, cs `cp1´Gq˘ ´1 1´G ˆcBGpr1´G, cs `cp1´Gqq ´1

1´GcBG`

r1´G, cs `cp1´Gq˘ ´1 1´G

ı

“Tr_Y´2

„

cBGδc 1

1´GcBGδc 1

1´GcBGδc 1 1´G

ȷ

“Tr_Y´2

„

cBδc G

1´Gδc G

1´Gδc G 1´G

ȷ

(D.5)

“Tr_Y´2rcBδcpG`KqΩ<sub>x</sub>δcpG`KqΩ_xδcpG`KqΩ_xs

“Tr_Y´2rcBδcGΩ_xδcGΩ_xδcGΩ_xs (D.6)

`Tr_Y´2rcBδcGΩ_xδcKΩ_xδcKΩ_xs (D.7)

`Tr_Y´2rcBδcKΩ_xδcGΩ_xδcKΩ_xs (D.8)

`Tr_Y´2rcBδcKΩ_xδcKΩ_xδcGΩ_xs. (D.9) The first term (D.6) becomes

(D.6)“Tr_Y´2rcBδcΩ_xpBc´δcGqGΩ_xδcGΩ_xs

“ 1

2Tr_Y´2rδpcBδcΩ_xBcΩ_xBcΩ_xqs ´Tr_Y´2rcBδcΩ_xBcΩ_xδcKΩ_xs

´ 1

2Tr_Y´2rδpcBδcΩ_xδcKΩ_xδcΩ_xqs

“ ´1

2Tr_Y´2rcBδcΩ_xBpδcqΩ_xBcΩ_xs ´1

2Tr_Y´2rcBδcΩ_xBcΩ_xBpδcqΩ_xs

`Tr_Y´2rBcδcΩ_xBcΩ_xδcKΩ_xs ´ 1

2Tr_Y´2rcBpBcqΩ_xδcKΩ_xδcΩ_xs

` 1

2Tr_Y´2rcBδcΩ_xpBcqKΩ_xδcΩ_xs ´ 1

2Tr_Y´2rcBδcΩ_xδcKΩ_xpBcqΩ_xs

“6

¡8

0

dx₁dx₂dx₃e^´px1`x2`x3qlim

yÑ0ByBcdggry`x<sub>1</sub>;x<sub>1</sub>`x₂`x<sub>3</sub>`ys

“ ´6pπ²´6q

π⁴ . (D.10)

The second term (D.7) becomes (D.7)“ 1

2TrY´2rδpδcKΩxδcKΩxcBδcΩxqs

“ 1

Tr rpBcqKΩ δcKΩ cBδcΩ s

´1

2Tr_Y´2rδcKΩ_xpBcqKΩ_xcBδcΩ_xs

`1

2Tr_Y´2rδcKΩ_xδcKΩ_xcBpBcqΩ_xs

“

¡8

0

dx1dx2dx3e^´px1`x2`x3q lim

y,y¹Ñ0ByBy¹

ˆ

!

´4¨Bcdggry`x<sub>1</sub>;x<sub>1</sub>`x₂`x<sub>3</sub> `y`y¹s

`2¨Bcdggrx1;x1`x2 `x3`y`y¹s )

“ ´24´2π²

π⁴ ´π²´12 π⁴

“ ´12´π²

π⁴ . (D.11)

We can show that the remaining terms (D.8) and (D.9) are equal to (D.7):

(D.7)“(D.8)“(D.9). (D.12)

Therefore,

(D.2)“(D.10)`3ˆ(D.7)“ ´ 3

π². (D.13)

We obtain the energy of the half-brane solution by adding (D.1) and (D.2):

1

6TrY´2rΨ_1{2³s “ 1 6

ˆ 3 2π² ´ 3

π²

˙

“ ´ 1

4π² “EpΨ0q ` 1

2T9. (D.14)

Appendix E

Detailed Calculations of the EOMS for Ψ _3{2

We give the detailed calculations of the remaining terms of the EOMS for Ψ_3{2. The term (5.2.14) becomes

(5.2.14)“lim

ϵÑ0ϵTrY´2

„

Bγ²c 1

1´G_ϵBc 1

´G_ϵ ȷ

“lim

ϵÑ0ϵTrY´2

„

Bγ²cGϵΩϵBcGϵ

1

´K_ϵ ȷ

“lim

ϵÑ0

ϵ 2TrY´2

„ δ

ˆ

Bγ²cΩϵBpδcq 1

´K_ϵ

˙ȷ

“lim

ϵÑ0

ˆϵ 2Tr_Y´2

„

BpBcγqcΩ_ϵBpδcq 1

´K_ϵ ȷ

` ϵ 2Tr_Y´2

„

Bγ²cΩ_ϵB²c 1

´K_ϵ ȷ˙

“ ´lim

ϵÑ0ϵTr_Y´2

„

BcBcΩ_ϵBγ 1

´K_ϵγ ȷ

“ ´lim

ϵÑ0ϵ¨lim

yÑ0By

␣Bcdggrz₁;x₁`y`z₁s ´Bcdggrz₁`y;x<sub>1</sub>`y`z₁s(

“(5.2.20)

“0, (E.1)

where Ω_ϵ :“ _1´K¹

ϵ, and other notations are explained shortly. Therefore, the first term (5.2.10) vanishes:

(5.2.10)“(5.2.20)`(E.1)“0. (E.2) For simplicity, we use the certain letters x_i and z_i, as Schwinger parameters corre-sponding to the following Laplace transformations:

1 1´K_ϵ “

ż8

0

dx_ie^´p1`ϵqxiΩ^xi, 1

´K_ϵ “ ż8

0

dz_ie^´ϵziΩ^zi. (E.3)

In the following, we omitş8

0 dxi andş8

0 dzi and also the exponential factors. For example,

we abbreviate the term:

Tr_Y´2

„

BcBcΩ_ϵBγ 1

´Kϵ

γ ȷ

“ lim

yÑ0By

ij8

0

dx₁dz₁e^´p1`ϵqx1e^´ϵz1

ˆ tBcdggrz1;x1`y`z1s ´Bcdggrz1`y;x1`y`z1su, (E.4) as

yÑ0limBytBcdggrz₁;x₁`y`z₁s¹ ´Bcdggrz₁`y;x<sub>1</sub>`y`z₁s¹u. (E.5) The second term (5.2.11) becomes

(5.2.11)“lim

ϵÑ0ϵTr_Y´2

„

p1´Bcq G_ϵK_ϵ 1´Gϵ

c 1

´Gϵ

γ² 1

´Gϵ

ȷ

“ ´lim

ϵÑ0ϵTr_Y´2

„

Bγ² 1

´Gϵ

c G_ϵK_ϵ 1´Gϵ

c 1

´Gϵ

ȷ

`lim

ϵÑ0ϵTr_Y´2

„ G_ϵK_ϵ 1´Gϵ

c 1

´Gϵ

γ² 1

´Gϵ

ȷ

“(5.2.10)´lim

ϵÑ0ϵTr_Y´2

„ K_ϵ 1´Gϵ

c 1

´Gϵ

γ² ȷ

“ ´lim

ϵÑ0ϵTr_Y´2

„

K_ϵG_ϵΩ_ϵcG_ϵ 1

´Kϵ

γ² ȷ

“ ´lim

ϵÑ0ϵTr_Y´2

„

G_ϵΩ_ϵBcG_ϵ 1

´Kϵ

γ² ȷ

“0. (E.6)

Here, we use the following equation which holds for the string field φ anti-commuting with B:

Tr_Y´2rφs “Tr_Y´2rpBc`cBqφs

“Tr_Y´2rBcφs `Tr_Y´2rcBφs

“TtY´2rBcφs ´TrY´2rBcφs

“0. (E.7)

The third term (5.2.12) becomes (5.2.12)“ ´lim

ϵÑ0ϵTr_Y´2

„

cB G_ϵK_ϵ 1´G_ϵc 1

´G_ϵc G_ϵK_ϵ 1´G_ϵc 1

´G_ϵ ȷ

“ ´lim

ϵÑ0ϵTr_Y´2

„

cB K_ϵ

1´G_ϵδc 1

´G_ϵc K_ϵ

1´G_ϵδc 1

´G_ϵ ȷ

“ ´lim

ϵÑ0ϵTr_Y´2

„

p´Bc`K_ϵqB 1

1´G_ϵδc 1

´G_ϵp´Bc`K_ϵcq 1

1´G_ϵδc 1

´G_ϵ ȷ

“lim

ϵÑ0ϵTr_Y´2

„

BcB 1 1´Gϵ

δc 1

´Gϵ

K_ϵc 1 1´Gϵ

δc 1

´Gϵ

ȷ

`lim

ϵÑ0ϵTr_Y´2

„

K_ϵcB 1

1´G_ϵδc 1

´G_ϵBc 1

1´G_ϵδc 1

´G_ϵ ȷ

“lim

ϵÑ02ϵTr_Y´2

„

Bc 1

1´G_ϵδc 1

´G_ϵBc 1

1´G_ϵδcG_ϵ ȷ

“lim

ϵÑ02ϵTr_Y´2

„

Bcp1`G_ϵqΩ_ϵδcG_ϵ 1

´K_ϵBcp1`G_ϵqΩ_ϵδcG_ϵ ȷ

“lim

ϵÑ02ϵTr_Y´2

„

BcΩ_ϵδcG_ϵ 1

´K_ϵBcΩ_ϵδcG_ϵ ȷ

(E.8)

`lim

ϵÑ02ϵTr_Y´2

„

BcG_ϵΩ_ϵδcG_ϵ 1

´K_ϵBcG_ϵΩ_ϵδcG_ϵ ȷ

. (E.9)

The terms (E.8) and (E.9) vanish as follows:

(E.8)“lim

ϵÑ02ϵTrY´2

„

BcΩϵGϵδcGϵ

1

´K_ϵBcΩϵδc ȷ

“lim

ϵÑ02ϵTrY´2

„

BcΩϵpBc´δcGϵqGϵ

1

´K_ϵBcΩϵδc ȷ

“lim

ϵÑ0ϵTrY´2

„ δ

ˆ

BcΩϵδcBcΩϵBc 1

´K_ϵ

˙ȷ

“lim

ϵÑ0

ˆ ϵTrY´2

„

BpδcqΩϵδcBcΩϵBc 1

´K_ϵ ȷ

´ϵTrY´2

„

BcΩϵδcBcΩϵBpδcq 1

´K_ϵ ȷ˙

“4 lim

ϵÑ0

ˆ ϵTrY´2

„

BcBc 1

´K_ϵBγΩϵγΩϵ

ȷ

´ϵTrY´2

„

BcBcΩϵγΩϵBγ 1

´K_ϵ ȷ˙

“8 lim

ϵÑ0lim

yÑ0BytBcdggrx₁;z₁`y`x₁`x<sub>2</sub>s<sup>1</sup> ´Bcdggrx<sub>1</sub>`y;z₁`y`x₁`x₂s¹u

“ ´lim

ϵÑ0ϵ ż8

0

da4ap2a²´π²pe^a´1qqe<sup>´apϵ`1q</sup> π<sup>2</sup>pa<sup>2</sup>`π²q

“ ´lim

ϵÑ04ϵpCipπϵqcospπϵq ` ¨ ¨ ¨ q

“0, (E.10)

and

(E.9)“lim

ϵÑ02ϵTr_Y´2

„

BcΩ_ϵδcG_ϵ 1

´Kϵ

BcG_ϵΩ_ϵδcK_ϵ ȷ

“ ´lim

ϵÑ02ϵTr_Y´2

„

BcΩ_ϵδcG_ϵ 1

´Kϵ

BpδcqΩ_ϵδcK_ϵ ȷ

“lim

ϵÑ02ϵTr_Y´2

„

BcΩ_ϵδcG_ϵ 1

´Kϵ

δcK_ϵΩ_ϵδcK_ϵ ȷ

“lim

ϵÑ0ϵTr_Y´2

„ δ

ˆ

δcK_ϵΩ_ϵδcK_ϵBcΩ_ϵδc 1

´Kϵ

˙ȷ

“lim

ϵÑ0

ˆ ϵTr_Y´2

„

pBcqK_ϵΩ_ϵδcK_ϵBcΩ_ϵδc 1

´Kϵ

ȷ

´ϵTr_Y´2

„

δcK_ϵΩ_ϵpBcqK_ϵBcΩ_ϵδc 1

´Kϵ

ȷ

`ϵTr_Y´2

„

δcK_ϵΩ_ϵδcK_ϵBcΩ_ϵpBcq 1

´K ȷ ˙

“lim

ϵÑ0

ˆ

4ϵTr_Y´2

„

BcBcK_ϵΩ_ϵγΩ_ϵBγ 1

´K_ϵ ȷ

`4ϵTr_Y´2

„

BcBcΩ_ϵBγ 1

´K_ϵBγΩ_ϵ ȷ

´4ϵTr_Y´2

„

BcBc 1

´K_ϵBγΩ_ϵγK_ϵΩ_ϵ ȷ ˙

“lim

ϵÑ04ϵTr_Y´2

„

BcBcΩ_ϵBγ 1

´K_ϵBγΩ_ϵ ȷ

“lim

ϵÑ04ϵ lim

y,y¹Ñ0ByBy¹

!

Bcdggrz₁`y<sup>1</sup>;x<sub>1</sub>`y`z<sub>1</sub>`y¹`x₂s¹

´Bcdggrz₁;x₁`y`z₁`y<sup>1</sup> `x₂s¹ ´Bcdggry`z<sub>1</sub>`y¹;x₁`y`z₁`y<sup>1</sup>`x₂s¹

`Bcdggry`z₁;x₁`y`z₁`y<sup>1</sup>`x₂s¹ )

“lim

ϵÑ0ϵ ż8

0

da2apa³` pe<sup>a</sup>`1qa²`π<sup>2</sup>a´π<sup>2</sup>pe<sup>a</sup>`1qqe<sup>´apϵ`1q</sup> pa<sup>2</sup>`π²q²

“ ´lim

ϵÑ02ϵpCipπϵqcospπϵq ` ¨ ¨ ¨ q

“0. (E.11)

Then we obtain

limϵÑ0EOMSpJΨ_3{2Kϵq „lim

ϵÑ0ϵˆ`

logϵ`Opϵ⁰q˘

“0. (E.12)

The solution lim_ϵÑ0JΨ_3{2Kϵ satisfies the EOMS.

Appendix F

Detailed Calculation of the Energy of the Tachyon Vacuum Solution in Berkovits’ SFT

We give detailed calculation of the energy of the tachyon vacuum solution in Berkovits’

SFT. The energy is given by Epg₀q “ ´Spg₀q “

ż1

0

dtTrrη₀`

g₀ptq^´1Btg₀ptq˘

¨g₀ptq^´1Qg₀ptqs. (F.1) We can rewrite the integrand:

Trrη₀`

g₀ptq^´1Btg₀ptq˘

¨g₀ptq^´1Qg₀ptqs

“Trrη₀`

pv₀u₀q^´1Btpv₀u₀q˘

pv₀u₀q^´1Qpv₀u₀qs

“Trrη₀pv^´1₀ Btv₀qpv₀^´1Qv₀ `Qu₀¨u₀^´1qs

“Trrη₀pv˜₀Btv₀qv˜₀Qv₀s. (F.2) Here we used g₀ptq “v₀u₀,Btu₀ “0,v₀^´1 “v˜₀`Bcand the fact that the string fields u₀, u₀^´1, Qu₀¨u₀^´1, BcB_tv₀, and BcQv₀ PH^small. We write down v₀ expicitly:

v₀ “

„1 ´α α t¯¨I´K ´αV

ȷ

“1`αζ ´αγB`cp¯t¨I´K´αV ´1qB

“1`αζ ´αQζ ¨B `cp¯t¨I´K´1qB. (F.3) We perform Q forv₀:

Qv₀ “αQζ`αQζK`Qc¨ p¯t¨I´K´1qB´cpt¯¨I´K ´1qK

“cpαVp1`Kq ` Bcp¯t¨I´K´1qB ´ p¯t¨I ´K´1qKq

`Bγ<sup>2</sup>p¯t¨I´K´1q `αγp1`Kq

“c`

´det₀p1`Kq ` Bcp¯t¨I´K´1qB`t¯¨I`α²p1`Kq˘

`Bγ<sup>2</sup>p¯t¨I´K´1q `αγp1`Kq. (F.4) We perform Bt forv₀:

We decompose it as Btv₀ “v9^l`v9^s₀, depending on whether it is in H^small or not.

v9^l :“qζ´qcV B RH^small, (F.6) v9₀^s :“ ´qγB´cIB PH^small. (F.7) We write down the explicit form of w and decompose it

w“

„´α² α

´α 1 ȷ

“ ´α²Bc´αζ `αγB`cB, (F.8)

w^l:“ ´αζ, (F.9)

w^s:“ ´α²Bc`αγB`cB. (F.10)

We want to calculate v˜₀Qv₀, and for that purpose we first calculate wQv₀: wQv₀ “`

p´α²B `αγ<sup>´1</sup>qc` pαγ `cqB˘

ˆ pcϕ₀`Bγ<sup>2</sup>p¯t¨I´K´1q `αγp1`Kqq

“ ´α²Bγ²p¯t¨I´K´1q `αcBγp¯t¨I´K´1q ´α<sup>3</sup>Bcγp1`Kq

´α²cp1`Kq `αγBcϕ₀`cϕ<sub>0</sub>`α²γBγp1`Kq `αcBγp1`Kq

“αγBp¯tαγ¨I´α²cp1`Kq `cϕ₀q

`cp¯tαBγ¨I´α<sup>2</sup>p1`Kq `ϕ0q. (F.11)

Here we defined

ϕ0 :“ ´det0p1`Kq ` Bcp¯t¨I´K´1qB`¯t¨I`α²p1`Kq. (F.12) Because v˜₀Qv₀ “D₀wQv₀,

˜

v0Qv0 “ pγ 1

det₀Bζ `c 1 det₀Bq ˆ

´

αγBp¯tαγ¨I´α²cp1`Kq `cϕ₀q

`cp¯tαBγ¨I´α<sup>2</sup>p1`Kq `ϕ₀q

¯

“ pαγ `cq 1

det₀Bp¯tαγ¨I´α²cp1`Kq `cϕ₀q

“ pαγ `cq 1 det₀B´

¯tαγ¨I´α²cp1`Kq

`c`

´det₀p1`Kq ` Bcpt¯¨I´K ´1qB`¯t¨I`α²p1`Kq˘¯

“ pαγ `cq 1 det0

B`¯tαγ¨I´cdet<sub>0</sub>p1`Kq `tc¯ ¨I´ Bcp¯t¨I´K´1q˘

“ pαγ `cq 1

det₀BΦ₀. (F.13)

Here, we defined

Φ0 :“¯tαγ¨I´cdet0p1`Kq `tc¯ ¨I´ Bcp¯t¨I´K´1q. (F.14) We rewrite (F.2) as

Trrη₀p˜v₀Btv₀q˜v₀Qv₀s “Trrη₀`

D₀pw^l`w<sup>s</sup>qpv9<sup>l</sup>`v9^s₀q˘

˜

v₀Qv₀s. (F.15) The explicit forms of the terms in pw^l`w<sup>s</sup>qpv9<sup>l</sup>`v9₀^sq are

w^lv9^l “ ´αζpqζ´qcV Bq “ 0, (F.16)

w^lv9₀^s “ ´αζp´qγB´cIBq “qαcB, (F.17) w^sv9^l “ p´α²Bc` pαγ `cqBqpqζ´qcV Bq “qpαγ`cqBpζ´Vq, (F.18) w<sup>s</sup>v9<sub>0</sub><sup>s</sup> “ p´α<sup>2</sup>Bc` pαγ `cqBqp´qγB´cIBq “ pqα²γ´αγI ´cIqB (F.19) Then, (F.15) becomes

(F.15)“Trrη₀D₀¨w^lv9₀^sv˜₀Qv₀s (F.20)

`Trrη₀pD₀w^sv9^lq˜v₀Qv₀s (F.21)

`Trrη0D0¨w^sv9₀^sv˜0Qv0s. (F.22) The factors η₀D₀ and η₀pD₀w^sv9^lq are calculated as follows:

η₀D₀ “η₀ ˆ

γ 1

det₀Bζ`c 1 det₀B

˙

“ ´γ ˆ

η₀ 1 det₀

˙

Bζ´γ 1

det₀Bcpη₀γ^´1q ´c ˆ

η₀ 1 det₀

˙

B, (F.23)

η₀pD₀w^sv9^lq “ η₀ ˆˆ

γ 1

det₀Bζ`c 1 det₀B

˙

qpαγ `cqBpζ´Vq

˙

“qη₀ ˆ

αγ 1

det₀Bpζ´Vq `c 1

det₀Bpζ´Vq

˙

“ ´qpαγ`cqη₀ ˆ 1

det₀Bpζ´Vq

˙

. (F.24)

The first term (F.20) becomes (F.20)“Trrη₀D₀¨w^lv9₀^sv˜₀Qv₀s

“ ´Tr

„ c

ˆ η₀ 1

det₀

˙

BqαcBpαγ`cq 1 det₀BΦ₀

ȷ

“qαTr

„ˆ η₀ 1

det₀

˙ 1

det₀BΦ₀c ȷ

“qαTr

„ ˆ η₀ 1

det₀

˙ 1 det₀B

ˆ p¯tαγ¨Ic´cdet₀p1`Kqc`tc¯ ¨Ic´ Bcp¯t¨I´K´1qcq ȷ

, (F.25) where we used the cyclicity of Tr¹, Trrφ₁φ₂s “ p´q^{ϵpφ1qϵpφ2q}Trrφ₂φ₁s. Here,

´cdet₀p1`Kqc`tc¯ ¨Ic´ Bcp¯t¨I´K´1qcq

“ ´prc,det₀s `det<sub>0</sub>cqp1`Kqc`¯tc¨Ic´ Bcpt¯¨I´K ´1qcq

“ ´prc,tI¯ ´K ´αV `α<sup>2</sup>s `det₀cqp1`Kqc`¯tc¨Ic´ Bcpt¯¨I´K ´1qcq

“ ´p¯trc, Is ` Bc`det₀cqp1`Kqc`¯tc¨Ic´ Bcp¯t¨I´K´1qcq

“ ´det₀cBc´t¯rc, Isp1`Kqc´ Bcp1`Kqc`¯tcIc´tBc¯ ¨Ic` BcpK `1qc

“ ´det₀cBc´t¯rc, IsKc´¯tBc¨Ic

“ ´det₀cBc´t¯pcIKc´IcBc` Bc¨Icq

“ ´det₀cBc`tpIcBc¯ ´KcIcq

“ ´det₀cBc`tpIcBc¯ ´KcrI, csq, (F.26)

where we used

rc,det₀s “¯trc, Is ` Bc. (F.27) Therefore,

Φ₀c“tαγ¯ ¨Ic´det₀cBc`¯tpIcBc´KcrI, csq. (F.28) We continue to calculate the term (F.25):

(F.25)“qαTr

„ˆ η₀ 1

det₀

˙ 1

det₀BΦ₀c ȷ

“qαTr

„ˆ η₀ 1

det₀

˙ 1 det₀B´

¯tαγ¨Ic´det₀cBc`¯t`

IcBc´KcrI, cs˘¯ȷ

“qαTr

„ˆ η₀ 1

det₀

˙ 1 det₀B´

¯tαγ¨Ic`¯t`

IcBc´KcrI, cs˘¯ȷ

. (F.29)

Here, we used the following equation:

Tr

„ˆ η₀ 1

det0

˙ 1 det0

Bdet₀cBc ȷ

“Tr

„ η₀

ˆ 1 det0

BcBc

˙ȷ

“0, (F.30)

where the first and the second equality comes from rB,det₀s “ 0 and Trrη₀φs “ 0, respectively.

The second term (F.21) becomes (F.21)“Trrη₀pD₀w^sv9^lqv˜₀Qv₀s

1In the case of the cubic theory, since the non-vanishing ghost number input in the trace is 3, then ϵpφ1qϵpφ2q “0. While, in the case of the Berkovits’ SFT, since the ghost number is 2, it may appear a minus sign in the cyclicty of the trace.

“ ´Tr

„

qpαγ`cqη₀ ˆ 1

det₀Bpζ´Vq

˙

pαγ`cq 1 det₀BΦ₀

ȷ

“ ´Tr

„

qpαγ`cqη₀ ˆ 1

det₀Bpζ´Vqpαγ`cq

˙ 1 det₀BΦ₀

ȷ

“qαTr

„ η₀

ˆ 1

det₀pα´Vq

˙ 1

det₀BΦ₀γ ȷ

(F.31)

`qTr

„ η₀

ˆ 1

det₀pα´Vq

˙ 1

det₀BΦ₀c ȷ

. (F.32)

The first term (F.31) becomes (F.31)“qαTr

„ η0

ˆ 1

det₀pα´Vq

˙ 1

det₀BΦ0γ ȷ

“qαTr

„ η0

ˆ 1

det₀pα´Vq

˙ 1 det₀B

ˆ`tαγ¯ ¨I´cdet0p1`Kq `¯tc¨I ´ Bcp¯t¨I´K´1q˘ γ

ȷ

“qαTr

„ η0

ˆ 1

det₀pα´Vq

˙ 1

det₀Bp´cdet0p1`Kq `¯tc¨Iqγ ȷ

“ ´qαTr

„ η0

ˆ 1

det₀pα´Vq

˙ 1

det₀Bprc,det0s `det0cqp1`Kqγ ȷ

`tqαTr¯

„ η0

ˆ 1

det₀pα´Vq

˙ 1

det₀Bc¨Iγ ȷ

“ ´qαTr

„ η₀

ˆ 1

det₀pα´Vq

˙ 1

det₀Bp¯trc, Is ` Bc`det₀cqp1`Kqγ ȷ

`tqαTr¯

„ η₀

ˆ 1

det₀pα´Vq

˙ 1

det₀Bc¨Iγ ȷ

“ ´qαTr

„ η₀

ˆ 1

det₀pα´VqBcp1`Kqγ

˙ȷ

`tqαTr¯

„ η₀

ˆ 1

det₀pα´Vq

˙ 1

det₀Bc¨Iγ ȷ

“ ´¯tqαTr

„ 1

det₀pα´Vq ˆ

η₀ 1 det₀

˙

Bc¨Iγ ȷ

, (F.33)

where we usedTrrBφs “0forφ s.t. rB, φu “0andTrrη₀φs “0. The second term (F.32) becomes

(F.32)“qTr

„ η₀

ˆ 1

det₀pα´Vq

˙ 1

det₀BΦ₀c ȷ

“qTr

„ η0

ˆ 1

det₀pα´Vq

˙ 1 det₀B

ˆ`¯tαγ¨Ic´det0cBc`tpIcBc¯ ´KcrI, csq˘ ȷ

“ ´qTr

„ η₀

ˆ 1

det₀pα´VqBcBc

˙ȷ

`qTr

„ η₀

ˆ 1

det₀pα´Vq

˙ 1 det₀B´

¯tαγ¨Ic`¯t`

IcBc´KcrI, cs˘¯ȷ

“ ´qTr

„ 1

det₀pα´Vq ˆ

η₀ 1 det₀

˙ B´

¯tαγ¨Ic`¯t`

IcBc´KcrI, cs˘¯ȷ

. (F.34) Then, we reach the following form of (F.21):

(F.21)“(F.33)`(F.34)

“ ´tqαTr¯

„ 1

det₀pα´Vq ˆ

η0

1 det₀

˙

Bc¨Iγ ȷ

´qTr

„ 1

det₀pα´Vq ˆ

η0

1 det₀

˙ B

´¯tαγ¨Ic`t¯`

IcBc´KcrI, cs˘¯ȷ

“ ´tqαTr¯

„ 1

det₀pα´Vq ˆ

η0

1 det₀

˙

rI, γsBc ȷ

´tqTr¯

„ 1

det₀pα´Vq ˆ

η0

1 det₀

˙ B`

IcBc´KcrI, cs˘ ȷ

. (F.35)

The third term (F.22) becomes (F.22)“Trrη₀D₀¨w^sv9₀^sv˜₀Qv₀s

“Tr

„

η₀D₀pqα²γ´αγI ´cIqBpαγ`cq 1 det₀BΦ₀

ȷ

“Tr

„

η₀D₀pqα²γ´αγI ´cIq 1 det₀BΦ₀

ȷ

“Tr

„ ˆ

´γ ˆ

η₀ 1 det₀

˙

Bζ ´γ 1

det₀Bcpη₀γ^´1q ´c ˆ

η₀ 1 det₀

˙ B

˙

ˆ pqα²γ´αγI´cIq 1 det₀BΦ₀

ȷ

“ ´Tr

„ γ

ˆ η₀ 1

det₀

˙

Bζpqα²γ´αγI´cIq 1 det₀BΦ₀

ȷ

(F.36)

´Tr

„ γ 1

det₀Bcpη₀γ^´1qpqα²γ´αγI´cIq 1 det₀BΦ₀

ȷ

(F.37)

´Tr

„ c

ˆ η₀ 1

det₀

˙

Bpqα²γ ´αγI ´cIq 1 det₀BΦ₀

ȷ

. (F.38)

The first term (F.36) becomes (F.36)“ ´Tr

„ γ

ˆ η₀ 1

det₀

˙

Bζpqα²γ´αγI´cIq 1 det₀BΦ₀

ȷ

“Tr

„ˆ η0

1 det₀

˙

pqα²´αIq 1

det₀BΦ0γ ȷ

“Tr

„ ˆ η₀ 1

det₀

˙

pqα²´αIq 1 det₀B

ˆ`¯tαγ¨I´cdet<sub>0</sub>p1`Kq `tc¯ ¨I´ Bcp¯t¨I´K ´1q˘ γ

ȷ

“Tr

„ˆ η₀ 1

det₀

˙

pqα²´αIq 1

det₀Bp´cdet₀p1`Kq `tc¯ ¨Iqγ ȷ

“Tr

„ˆ η₀ 1

det₀

˙

pqα²´αIq 1 det₀B´

´`

rc,det₀s `det₀c˘

p1`Kq `tc¯ ¨I¯ γ

ȷ

“¯tTr

„ˆ η₀ 1

det₀

˙

pqα²´αIq 1

det₀Bc¨Iγ ȷ

. (F.39)

The second term (F.37) vanishes (F.37)“ ´Tr

„ γ 1

det₀Bcpη0γ^´1qpqα²γ´αγI ´cIq 1 det₀BΦ0

ȷ

“ ´Tr

„ γ 1

det₀Bcpη0γ^´1qpqα²γ´αγIq 1 det₀BΦ0

ȷ

“ ´Tr

„ γ 1

det₀Bc`

η0pγ^´1γq˘

pqα²´αIq 1 det₀BΦ0

ȷ

“0. (F.40)

The third term (F.38) becomes (F.38)“ ´Tr

„ c

ˆ η0

1 det₀

˙

Bpqα²γ´αγI ´cIq 1 det₀BΦ0

ȷ

“ ´Tr

„ˆ η0

1 det₀

˙ I 1

det₀BΦ0c ȷ

“ ´Tr

„ˆ η0

1 det₀

˙ I 1

det₀B

´tαγ¯ ¨Ic´det0cBc`¯t`

IcBc´KcrI, cs˘¯ȷ

“ ´¯tTr

„ˆ η0

1 det₀

˙ I 1

det₀Bpαγ¨Ic`IcBc´KcrI, csq ȷ

. (F.41)

Then, we reach the following form of (F.22):

(F.22)“(F.39)`(F.40)`(F.41)

“¯tTr

„ˆ η₀ 1

det₀

˙

pqα² ´αIq 1

det₀Bc¨Iγ ȷ

´¯tTr

„ˆ η₀ 1

det₀

˙ I 1

det₀Bpαγ¨Ic`IcBc´KcrI, csq ȷ

“¯tqα²Tr

„ˆ η₀ 1

det₀

˙ 1

det₀Bc¨Iγ ȷ

´¯tαTr

„ˆ η₀ 1

det₀

˙ I 1

det₀rI, γsBc ȷ

´¯tTr

„ˆ η₀ 1

det₀

˙ I 1

det₀BpIcBc´KcrI, csq ȷ

. (F.42)

We summarize the above calculations:

Trrη₀`

g₀ptq^´1Btg₀ptq˘

g₀ptq^´1Qg₀ptqs

“qαTr

„ˆ η₀ 1

det₀

˙ 1 det₀B´

tαγ¯ ¨Ic`¯t`

IcBc´KcrI, cs˘¯ȷ

´¯tqαTr

„ 1

det₀pα´Vq ˆ

η₀ 1 det₀

˙

rI, γsBc ȷ

´¯tqTr

„ 1

det₀pα´Vq ˆ

η₀ 1 det₀

˙

BpIcBc´KcrI, csq ȷ

`¯tqα²Tr

„ˆ η₀ 1

det₀

˙ 1

det₀Bc¨Iγ ȷ

´¯tαTr

„ˆ η₀ 1

det₀

˙ I 1

det₀rI, γsBc ȷ

´¯tTr

„ˆ η₀ 1

det₀

˙ I 1

det₀BpIcBc´KcrI, csq ȷ

“¯t

# qα²Tr

„ˆ η₀ 1

det₀

˙ 1

det₀Bγ¨Ic ȷ

`qαTr

„ˆ η₀ 1

det₀

˙ 1

det₀BpIcBc´KcrI, csq ȷ

´qα²Tr

„ 1 det0

ˆ η₀ 1

det0

˙

rI, γsBc ȷ

`qαTr

„ 1 det0

V ˆ

η₀ 1 det0

˙

rI, γsBc ȷ

´qαTr

„ 1 det0

ˆ η₀ 1

det0

˙

BpIcBc´KcrI, csq ȷ

`qTr

„ 1 det0

V ˆ

η₀ 1 det0

˙

BpIcBc´KcrI, csq ȷ

`qα²Tr

„ˆ η₀ 1

det0

˙ 1 det0

Bc¨Iγ ȷ

´αTr

„ˆ η₀ 1

det0

˙ I 1

det0

rI, γsBc ȷ

´Tr

„ˆ η₀ 1

det₀

˙ I 1

det₀BpIcBc´KcrI, csq ȷ+

“¯t

#

2qα²Tr

„ˆ η0

1 det₀

˙ 1

det₀BcrI, γs ȷ

`2qαTr

„ˆ η₀ 1

det₀

˙ 1

det₀BpIcBc´KcrI, csq ȷ

`qαTr

„ 1 det₀V

ˆ η₀ 1

det₀

˙

rI, γsBc ȷ

`qTr

„ 1 det₀V

ˆ η₀ 1

det₀

˙

BpIcBc´KcrI, csq ȷ

´αTr

„ˆ η₀ 1

det₀

˙ I 1

det₀rI, γsBc ȷ